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Mathematicians use phrases like “it is obvious that”, “it is easy to see that”, “it is clear that”, and “is trivial” to indicate that some steps have been omitted. The use of such language may be classified under three headings — honest, dishonest, and pedagogical.

The honest use of these phrases occurs when only a few steps have been omitted and these steps are simple enough that the average reader can easily fill in the gaps. Omitting such steps can be beneficial because it cuts down the length of an exposition and keeps the main ideas from getting lost amidst a morass of boring details and routine operations. By reminding the reader of small omissions in an unobtrusive fashion, such phrases help put the reader at ease — if they are left out, it is easy for the reader to be thrown off-course by a missing step or be left with an uneasy feeling that there might be a hole in a proof.

The dishonest use of these phrases occurs when a somewhat lengthy calculation has been left out because the author was too lazy to write it down. By using these phrases, the author hopes to intimidate potential critics from pointing out that material is missing by insinuating that anyone who would point out that something is missing is too stupid to fill in a few obvious steps. Frequent dishonest use of these terms may be a symptom of mathematheosis.

The pedagogical use of these phrases occurs when the author has deliberately left the filling-in of missing steps as an exercise to the reader.

Comments

The term comes from Quine's "quiddities". He invented this term by analogy with such medical conditions such as psychosis and sclerosis, to describe mthemathical hubris. As Quine describes it, a victim of mathematheosis is someone who constantly, and somewhat obnoxiously shows off his mathematical superiority by overusing up-to-date mathematical terminology and giving the impression that, because of his superior mathematical intelligence, he is exempt from the drudgery of extensive calculations and proof to which lesser mortals are subject.

When I get some time, maybe I'll add a link to Quine's home page (written in memoriam by the logician's son).

I think your post above could be made into a nice Encyclopedia
entry. The page you referenced is http://www.wvquine.org; it does
not have the text for Quiddities, but links to Amazon where
its ISBN is given.

I agree that a definition of mathematheosis would make a nice edition to PM. Although the phrases ``obvious'', ``clear'', and ``easy to see'' are probably the most common for this phenomenon, the word ``trivial'' needs added to this list. I know a handful of people who say ``trivial'' when referring to mathematics that they think is easy but is in reality quite difficult. In fact, trivial is the word that I most often hear in reference to this ``exemption'' that rspuzio describes.

Some mathematicians prefer the phrase: "it is well known that" to indicate that some result is true but the proof would require extra work on the part of the author and that the proof should generally be accessible in standard references. This gives the author some
protection from the "dishonest" use of the phrase in that the author is not pretending that the result is simple minded. Instead "it is well known" can often be interpretted to mean "consulting standard sources it follows that..."

For example:

"It is well known that the trace of AB equals the trace of BA." This is an exhaustingly long computation to check (without appealing to characterisitic polynomials) but indeed you can usually find a proof in any linear algebra that introduces the trace of a matrix.

"It is well known that the only odd order finite simple groups are cyclic of prime power." This requires the Feit-Thompson theorem yet the result actually IS well known amongst people who know what finite simple groups mean.